For What Values Of $c$ Does the Curve Have Maximum and Minimum Points for the and Minimum

question 155

Multiple Choice

For what values of $c$ does the curve have maximum and minimum points for the given function $f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1 ?$
Select the correct answer.

• A) For \(c = 0 , f ( x ) = - 5 x ^ { 2 } + 1\), a parabola whose vertex \(( 0,2 )\), is the absolute maximum.
• For \(c > 0 , f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1\), opens downward with one minimum points.
• For \(c < 0\), the graph opens upward, and has an absolute maximum at \(x = 0\) and no local minimum.
• B) For \(c = 0 , f ( x ) = 5 x ^ { 2 } + 1\), a parabola whose vertex \(( 0,2 )\), is the absolute maximum.
• For \(c > 0 , f ( x ) = c x ^ { 4 } + 5 x ^ { 2 } + 1\), opens upward with two minimum points.
• For \(c < 0\), the graph opens downward, and has an absolute minimum at \(x = 0\) and no local minimum.
• C) For \(c = 0 , f ( x ) = 5 x ^ { 2 } + 1\), a parabola whose vertex \(( 0,3 )\), is the absolute maximum.
• For \(c > 0 , f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1\), opens upward with two minimum points.
• For \(c < 0\), the graph opens downward, and has an absolute minimum at \(x = 0\) and no local minimum.
• D) For \(c = 0 , f ( x ) = - 5 x ^ { 2 } + 1\), a parabola whose vertex \(( 0,1 )\), is the absolute maximum.
• For \(c > 0 , f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1\), opens upward with two minimum points.
• For \(c < 0\), the graph opens downward, and has an absolute maximum at \(x = 0\) and no local minimum.
• E) For \(c = 0 , f ( x ) = - 5 x ^ { 2 } + 1\), a parabola whose vertex \(( 0 , - 1 )\), is the absolute maximum.
• For \(c > 0 , f ( x ) = c x ^ { 4 } + 5 x ^ { 2 } - 1\), opens downward with two maximum points.
• For \(c < 0\), the graph opens upward, and has an absolute minimum at \(x = 0\).

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Definitions:

Conservation

A cognitive principle in child development discovered by Piaget, indicating the understanding that quantity doesn't change despite changes in the shape or appearance of objects.

Piagetian Principle

A concept from Jean Piaget's theory, indicating that cognitive development occurs through a series of stages, characterized by distinct ways of thinking and understanding the world.

Clay

A natural, earthen material composed of fine-grained minerals, used in arts and crafts for sculpting and pottery.

Conservation

In cognitive development, the understanding that certain properties of objects can remain the same despite changes in their form or appearance.